Performance of Under-resolved Two-DImensional Incompressible … · TWO-DIMENSIONAL INCOMPRESSIBLE...

JOURNAL OF COMPUTATIONAL PHYSICS 138, 734–765 (1997)ARTICLE NO. CP975843

Performance of Under-resolvedTwo-Dimensional Incompressible Flow

Simulations, II1

Michael L. Minion* and David L. Brown†

*Mathematics Department, University of North Carolina, Chapel Hill, North Carolina 27599;and †Computing, Information, and Communication Division, Los Alamos National Laboratory,

Los Alamos, New Mexico 87545E-mail: *[email protected] and †[email protected]

Received December 26, 1996

This paper presents a study of the behavior of several difference approxima-tions for the incompressible Navier–Stokes equations as a function of thecomputational mesh resolution. In particular, the under-resolved case is con-sidered. The methods considered include a Godunov projection method, aprimitive variable ENO method, an upwind vorticity stream-function method,centered difference methods of both a pressure–Poisson and vorticity stream-function formulation, and a pseudospectral method. It is demonstrated thatall these methods produce spurious, nonphysical vortices of the type describedby Brown and Minion for a Godunov projection method (J. Comput. Phys.121, 1995) when the flow is sufficiently under-resolved. The occurrence ofthese artifacts appears to be due to a nonlinear effect in which the truncationerror of the difference method initiates a vortex instability in the computedflow. The implications of this study for adaptive mesh refinement strategiesare also discussed. Q 1997 Academic Press

Key Words: incompressible flow; mesh refinement; numerical accuracy.

1. INTRODUCTION

Even with the most powerful computers available today, the ability to accuratelyand effectively model realistic engineering problems that include incompressiblefluid flow is limited by the memory size and total processor speed of the computer

1 This work supported by the U.S. Department of Energy through Contracts W-7405-ENG-36, DE-FG02-92ER25139, and DE-AC03-76SF00098.

734

0021-9991/97 $25.00Copyright 1997 by Academic PressAll rights of reproduction in any form reserved.

735TWO-DIMENSIONAL INCOMPRESSIBLE FLOW SIMULATIONS

being used. An inevitable consequence of this situation is the necessity for a scientistor engineer to be able to make computations with as little resolution as possiblewhile still maintaining sufficient accuracy to obtain reasonable results. For thisreason, it is important to understand the behavior of difference methods at thelimits of under-resolution. Simply put, an under-resolved computation is one inwhich the grid spacing of the computational mesh is too large to resolve the smallestphysically relevant scales of the solution. For the Navier–Stokes equations thesmallest physically relevant scale is determined by the magnitude of the physicalviscosity [17]. A surprising aspect of such computations is that the numerical solutionmay appear to be quite smooth relative to the computational mesh despite beinginadequately resolved.

Many modern difference methods are designed to compute solutions that do notexhibit mesh-scale oscillations or become catastrophically unstable regardless ofwhether the flow is being fully resolved. In addition, such methods attempt tocompute small-scale features such as shear layers with as close to the correct scalesas can be supported by the computational mesh. We characterize such methods asbeing robust. We contrast these methods both with nonrobust methods, whichare unable to resolve such features without producing mesh-scale oscillations andpossibly instability, and also with artificial viscosity methods that add enough diffu-sion to stabilize the computation on any mesh. Many of the latter methods stabilizethe computation at the expense of widening the small-scale features well beyondthe correct physical length scales. In this paper, we will only consider methods thatwe have categorized as ‘‘robust’’ and ‘‘nonrobust.’’

In an earlier paper [9], a study of the behavior of a Godunov projection methodfor the time-dependent incompressible Navier–Stokes equations on meshes thatprovided insufficient resolution was presented. This particular method is of therobust category mentioned above, and in that paper an attempt was made to evaluatethe reliability of under-resolved numerical solutions using this method. It was shownthat for the model problem of a doubly-periodic shear layer pair, spurious vorticestypically appear in under-resolved calculations. While these extra vortices mightappear to be physically reasonable, they disappear in mesh-refinement studies, thusjustifying their classification as ‘‘spurious.’’

The numerical experiments discussed in the earlier paper were restricted to theGodunov projection method, with a brief discussion of similar effects observed ina centered-difference approximation as well as reference to related results in theliterature. The present paper extends our previous work by making a more exhaus-tive study of a variety of difference methods of both upwind and centered-differencetype on this shear layers. While this model Rayleigh–Taylor problem certainly is notrepresentative of all possible flows described by the incompressible Navier–Stokesequations, it is one of the classic unstable problems of incompressible flow and isa good model for understanding mixing effects. It is a reasonable requirementfor any numerical method that it solve this problem with an appropriate degreeof accuracy.

As in the earlier paper, only the Navier–Stokes problem is considered here. Theinviscid, or Euler problem, is much more difficult to address since without theviscous length scale, there is no dimensionless number that determines a smallest

736 MINION AND BROWN

physical scale. For computations, then, the smallest scale will be determined by themesh size and the size of any computational dissipation in the numerical method.In this case, it is impossible to quantify the concept of under-resolution.

In Section 3 a brief description of the numerical methods used in the study ispresented. We again consider a Godunov projection method [9, 24, 25] and compareit to an ENO method for the primitive variable formulation of the incompressibleNavier–Stokes equations [12, 13] as well as an upwind vorticity stream-functionformulation [23]. We also consider two centered-difference approximations, Hen-shaw’s fourth-order method with hyperviscosity stabilization [16] and the centeredvorticity stream-function method used in [9]. Finally, we also study a pseudospec-tral method.

After presenting some convergence results in Section 4, we present calculationsthat further substantiate the speculation made in [9] that the appearance of spuriousvortices in under-resolved computations is not an artifact of any particular differenceapproximation, but occurs due to under-resolution with any of the methods, ofeither upwind or centered type. Methods with higher accuracy and which favorcentered differencing appear to have more resolution on coarser grids than lessaccurate methods, but with all of the methods studied, the breakdown in accuracyis manifest in the appearance of these extra vortices.

The test problem that we study consists of a doubly periodic shear to which asinusoidal perturbation perpendicular to the orientation of the shear layers is madeat the lowest wavenumber supported by the computational region. In the absenceof any additional perturbations, each of the shear layers rolls up in a single vortexas the flow evolves. If an additional perturbation is made at twice this wavenumber,then two vortices will appear on each layer; the appearance of this solution is quitesimilar to the under-resolved calculations with the spurious vortices. We thereforespeculate that the mechanism for the appearance of the spurious vortices is asfollows: because of the quadratic nonlinearity in the Navier–Stokes equations, thetruncation error for each of the difference methods is also a quadratic function ofhigher derivatives of the true solution. If the truncation error becomes too large,it therefore tends to perturb the solution at wavenumbers that are a multiple ofthe fundamental perturbation wavenumber for the simulation. The manifestationof this will be the initiation and growth of additional vortices in the computedsolution. We present some truncation error analysis as well as several numericalexamples supporting this conjecture in Section 5.

As we pointed out in the earlier paper [9], one of the most alarming aspects ofthese results is the fact that without careful mesh-convergence studies, it is difficultto tell the difference between features of the solution that are correct and thosethat are numerical artifacts. For example, comparing gross statistical measurementssuch as total energy or enstrophy usually is not sufficient to characterize under-resolution for two runs using a given method, one of which contains spuriousvortices, and the other of which does not. (They are, however, useful in characteriz-ing the magnitude of the diffusion in a computation, particularly when verifyingthat this diffusion mimics the true viscous effects.) Nevertheless, since all methodsappear to exhibit the same basic behavior, it is important to be able to understandhow to use these methods in real engineering applications. Since sufficient resolution


in regions where physical flow instabilities can occur is the key to accurate computa-tions, it is necessary to be able to detect when and where a flow is not being resolvedin a given computation. This issue is of paramount importance in the design ofadaptive mesh refinement (AMR) algorithms for incompressible flow [1, 25] inwhich the computational mesh is locally refined as more resolution is required. Thisapproach is briefly discussed in Section 7, and an example is provided demonstratingthat even AMR must be implemented prudently to obtain accurate results.

2. THE NAVIER–STOKES EQUATIONS

2.1. Primitive Variable Formulation

There are many formulations of the two-dimensional incompressible Navier–Stokes equations, and several of them are used in this paper. The most commonform is the primitive variable formulation, which can be written as

Ut 1 (U ? =)U 1 =p 5 nDU (1)

= ? U 5 0, (2)

where U 5 (u, v)T 5 U(x, y, t) is the fluid velocity vector with components u andv, the horizontal and vertical velocity, respectively; p 5 p(x, y, t) is the fluid pressure;n is the (assumed constant) kinematic viscosity; and subscripts denote partial differ-entiation. This version (and all other versions) of the equations are considered herein the spatially doubly periodic domain [0, 1] 3 [0, 1]. For this domain, no boundaryconditions are needed other than the requirement that the solutions be doubly pe-riodic.

2.2. Pressure–Poisson Formulation

The pressure–Poisson form of the Navier–Stokes equations replaces the diver-gence constraint (2) with an elliptic equation for the pressure. The pressure equationis derived by taking the divergence of the momentum equation (1) and eliminatingterms using the divergence constraint (2):

Dp 1 Ux ? =u 1 Uy ? =v 5 0. (3)

Together with the momentum equation (1), this is an equivalent formulation ofthe incompressible Navier–Stokes equations in a doubly periodic domain.

2.3 Projection Formulation

‘‘Projection’’ formulations of the incompressible Navier–Stokes equations canbe written using the Hodge decomposition theorem, which states that any vectorfield W can be decomposed into two parts, one of which is divergence-free and the


other solenoidal. Moreover, these components are orthogonal (see, e.g., Chorin[10]),

W 5 U 1 =f, = ? U 5 0, U 5 PW, =f 5 (I 2 P)W,

where P is the implied projection operator that decomposes the field. Note thatP2 5 P. Using P, the momentum equation can be written as

Ut 5 Ph2(U ? =)U 1 nDU j(4)=p 5 (I 2 P)h2(U ? =)U 1 nDU j.

In practice, the projection is implemented by solving an elliptic equation for f

formed by taking the divergence of the Hodge decomposition

Df 5 = ? W.

Once f is calculated, =f is subtracted from the non-divergence-free velocity W toobtain the divergence-free velocity U.

2.4. Rotational Projection Formulation

It is also possible to formulate a projection form of the equations in terms of thevorticity g 5 = 3 U, by noting the identity

g 3 U 5 (U ? =)U 212

=uUu2,

whence (4) becomes

Ut 5 Ph2g 3 U 1 nDU j (5)

=(p 112

uUu2 ) 5 (I 2 P)h2g 3 U 1 nDU j.

2.5. Vorticity Stream-Function Formulation

The vorticity stream-function formulation uses the vorticity g and stream functionc as dependent variables, where the stream function satisfies the Cauchy–Riemann equations,

(u, v) 5 (cy , 2cx ).

In these variables, the incompressible Navier–Stokes equations are given by

gt 5 2(ug)x 2 (vg)y 1 nDg (6)

Dc 5 2g. (7)


3. NUMERICAL METHODS

Here an overview of the numerical methods used in our study is presented. Ineach case only a brief description of the details of the method is given since furtherdetails are available in the References. Also included is a discussion of the intendedadvantages of each method as well as inherent disadvantages.

3.1. The Godunov Projection Method

In our original study [9], the numerical method used to study the effects of under-resolution for incompressible flows was the Godunov projection method describedin [5] which is based on the method of Bell et al. [4]. In this method the momentumequation (1) is discretized using the second-order Crank–Nicolson type formulation

Un11 2 Un

Dt1 =pn11/2 5 2[(U ? =)U]n11/2 1

n

2D(Un 1 Un11), (8)

where the term [(U ? =)U ]n11/2 is calculated explicitly using a second-order Godunovprocedure. Equation (8) in turn is approximated in a two-step procedure where anintermediate value U* is first computed by solving

U* 2 Un

Dt1 =p n11/2 5 2[(U ? =)U]n11/2 1

n

2D(Un 1 U*). (9)

Here =pn11/2 is an approximation to the time-centered pressure gradient. Once U*has been computed, values of Un11 and =pn11/2 are computed by using a discreteprojection P,

Un11 5 P(U*)

=pn11/2 5 =pn11/2 1 (I 2 P)(U*)/Dt.

The Godunov projection method used in the present paper is implemented withinan adaptive mesh refinement framework (although for the single-grid problemsnone of this capability is utilized). There are a few differences between this methodand the one used in [9] which originated either as improvements over the formeralgorithm or as modifications that facilitated the AMR implementation. Some addi-tional details of the AMR method are included in Section 7; a more completediscussion can be found in [25].

One major difference is that an approximate projection operator for whichP2 ? P is used in place of the exact discrete projection in [9]. Approximate projectionmethods have been introduced in a number of recent papers [2, 19, 25], and theyare motivated by the difficulty in implementing an accurate exact projection onlocally refined grids as well as difficulties associated with the noncompact natureof discrete ‘‘exact’’ projection operators. For a single periodic grid, the approximateprojection differs from the projection in [9] only in that the discrete Laplacian used


in the projection Poisson problem uses the standard five-point stencil as opposedto a decoupled five-point stencil.

The second difference concerns the form of the advective derivative terms. Theversion here follows [24] which presents a modification to the somewhat complicatedform of these terms which improves the stability range of the overall method. Thespecifics will not be presented here.

Finally, the term =p n11/2 appearing in Eq. (9) is computed differently in thecurrent version. In the original, the pressure term was simply lagged, i.e.,=p n11/2 5 =pn21/2. Here a more accurate value is computed by performing anadditional projection. Specifically,

=p n11/2 5 (I 2 P)(U n 1 Dt(2[(U ? =)U ]n11/2 2 =pn21/2 1 nDUn)).

It should be noted that although these modifications are not trivial, the two versionsof the method behave nearly identically on the test problems presented here.

Despite the fact that the method is used here on under-resolved problems withsteep gradients in the velocities, no form of slope limiter in the computation of theadvective derivatives (as described in the original method) was used, nor was anyfiltering or smoothing of the flow employed. The combination of the Crank–Nicolson form for viscous terms and the Godunov method for advection is sufficientto make this method robust to under-resolution without the addition of eitherlimiters or filtering.

3.2. The Pseudospectral Method

Since the problems considered here are set in a doubly periodic domain, it ispossible to solve the equations of motion in spectral space. It is convenient torewrite the rotational form of the momentum equation (5) as

Ut 5 A(U) 1 nDU, (10)

where

A(U) 5 P(2g 3 U).

The Laplacian term is divergence free in spectral space and hence is omitted fromthe projection.

Denoting the two-dimensional Fourier transform of a function G by G, it ispossible to transform (10) into a system of ordinary differential equations

(11)Ut 5 A(U`

) 2 nk2U,

where k denotes the Fourier wavenumber. The diffusive term is incorporateddirectly into the ODE through the use of an integrating factor. Define C 5

enk2tU and note that

(12)Ct 5 enk2t A(U`

).


The A(U`

) term is computed by first forming the vorticity g in spectral spacefrom u and v. Then the inverse FFT is applied and g 3 U is computed in realspace. Finally, this quantity is transformed back to spectral space where the projec-tion operator is applied. A standard fourth-order Runge–Kutta method is appliedto (12) to integrate (11), which after some work gives

U1 5 e2nk2Dt/2(Un 1

Dt2

A(U`

n ))

U2 5 e2nk2Dt/2Un 1

Dt2

A(U`

1 )

U3 5 e2nk2DtUn 1 e2nk2

Dt/2DtA(U`

2 )

Un11 5 e2nk2DtUn 1

Dt6

[e2nk2DtA(U`

n ) 1 2e2nk2Dt/2(A(U

`

1) 1 A(U`

2)) 1 A(U`

3 )].

No dealiasing or spectral filtering is done for this method. Consequently, the methodis not robust and can become catastrophically unstable when under-resolving flows.Spectral methods have the advantage of being more accurate than other finitedifference methods but are not applicable to many engineering problems wherecomplex geometries and/or nonuniform meshes are required.

3.3. The Pressure–Poisson Method

This method is adapted from the method for overlapping grids presented byHenshaw [16]. It is based on the pressure–Poisson formulation of the Navier–Stokesequations (1) and (3).

The right-hand side of the momentum equation (1) is discretized using fourth-order centered differences to replace each derivative term. The pressure equation(3) is solved in Fourier space using the spectral representation of the fourth-ordercentered-difference approximation to the Laplacian. The right-hand side of thepressure equation is also approximated using fourth-order centered differences.

As formulated above, this method contains no artificial diffusion and hence isnot robust. The addition of a hyperviscosity term to the numerical method is oneway in which the method can be stabilized. For the simulations in Section 5.3, asecond-order discretization of 2Uxxxx 2 Uyyyy scaled by 0.15h4 is added explicitlyto the physical viscosity terms to damp small scale oscillations in the flow. Notethat hyperviscosity is used here only for runs which are badly under-resolved andnot in Sections 4 or 5.1.

3.4. The ENO method

This method is taken from [12] where it is used in a similar study of the resolutionof the method on shear layer flows. It is based on the projection form of theNavier–Stokes equations (4) and uses fourth-order ENO differencing [27, 28] tocompute the derivative terms. Viscous terms are computed explicitly using fourth-order centered differences, and the temporal integration is done using a third-orderTVD Runge–Kutta method [28].


The projection operator is implemented in spectral space using a slightly differentbut mathematically equivalent form than that in Section 2.3. Instead of solving aPoisson problem for the gradient piece of the flow, the incompressible velocity isrecovered by solving the vorticity stream-function equation (7). Following [12], letP(u9, v9) 5 (u, v). Denoting the Fourier transform of u and v as u and v, theprojection in Fourier space is

u 5dk(dk u9 2 dj v9)

d 2j 1 d2

k

v 52dj(dk u9 2 dj v9)

d 2j 1 d2

k

where dj and dk are the spectral representations of the fourth-order x and y derivativeoperator. These are computed by taking the square root of the spectral form ofthe fourth-order difference approximation to the second derivative, for example,

dk 5 Ï(230 1 32 cos(2fkh) 2 2 cos(4fkh))/12h2.

This somewhat unusual form of the spectral derivative ensures that P2 5 P.Before the velocities are transformed back to real space, an 8th-order spectral

filter is applied to remove small-scale oscillations from the flow. Each Fouriercomponent is multiplied by

sj,k 5 e2a(m( j)81m(k)8),

where

m(i) 5 50 if uiu , k0

uiu 2 k0

N /2 2 k0else

,

and N is the number of terms in the Fourier series. For this method a 5 15 log(10),and k0 5 4 [18, 22]. This is a slightly different filter than appears in [12], but wasthe form implemented in a version of the code kindly furnished by the authors of[12] and subsequently used to verify our implementation. The combination ofspectral filtering and the ENO differentiation make this method robust to under-resolution.

3.5. The Centered Vorticity Stream-Function Method

This method is the same as was used in [9]. It is based on the vorticity stream-function formulation of the Navier–Stokes equations. The numerical method dis-cretizes the momentum equation (6) by replacing all derivatives on the right-handside by fourth-order centered differences. The stream function is computed from


the vorticity in Fourier space. Denoting the discrete Fourier coefficients of thevorticity by gi, j , the stream function is given by

c(x, y) 51

4f 2 ON/2

j,k52N/211

gj,k

j 2 1 k2 e2f i( jx1ky).

The velocities are computed from c using fourth-order centered differences, and thetemporal integration is done using a standard fourth-order Runge–Kutta method.

The above formulation produces a simple method that is fourth-order accuratein time and space. However, because it contains no artificial diffusion, it is not robust.

3.6. The Upwind Vorticity Stream-Function Method

This method is similar in nature to the centered vorticity stream-function methoddescribed in the last section, but uses a non-centered differencing for the advectivederivatives and is of lower-order accuracy in both space and time.

The specific details of the procedure for calculating the advective derivatives areas follows. (See [23] for a more complete discussion.) The stream function c iscalculated from the vorticity g by solving a discretized version of Eq. (7) using thestandard five-point discretization of the Laplacian denoted by D5 . Specifically,

D5ci, j 524ci, j 1 ci, j11 1 ci, j21 1 ci11, j 1 ci21, j

h2 5 2gi, j ,

where ci, j P c(ih, jh) indicate cell-centered values of the approximate streamfunction. Cell-staggered velocities are then defined from c,

ui, j11/2 5 (ci, j11 2 ci, j )/h(13)

vi11/2, j 5 2(ci11, j 2 ci, j )/h,

and consequently

gi, j 5 (vi11/2, j 2 vi21/2, j 2 ui, j11/2 1 ui, j21/2 )/h.

Using the velocities (13) and terminology from [15], MAC-staggered velocitiesare defined by

ui11/2, j 5 (ui, j11/2 1 ui11, j11/2 1 ui, j21/2 1 ui11, j21/2 )/4 (14)

vi, j11/2 5 (vi11/2, j 1 vi21/2, j 1 vi11/2, j11 1 vi21/2, j11)/4. (15)

It is easy to see that these velocities have MAC divergence zero, i.e.,

ui11/2, j 2 ui21/2, j 1 vi, j11/2 2 vi, j21/2

h5 0.


MAC-staggered values of the vorticity are then interpolated from cell-centeredvalues using an ENO-type procedure to determine the interpolation stencil. Specifi-cally, the first point in the stencil is chosen by upwinding, and the remaining pointsare chosen by the smoothness of the surrounding points. Once these values arecomputed, the advective derivatives are calculated conservatively,

A(g)i, j 5 (ui11/2, j gi11/2, j 2 ui21/2, j gi21/2, j 1 vi, j11/2gi, j11/2 2 vi, j21/2gi, j21/2 )/h.

The viscous term is incorporated explicitly and calculated using D5 . Hence thespatially discretized form of (6), (7) is

tgi, j 5 2A(g)i, j 1 nD5gi, j .

The temporal integration is performed using the same third-order TVD Runge–Kutta method as in the primitive variable ENO method above.

This method is taken from [23], where it was used to model the evolution ofvortex patches and hence possesses two qualities that are desirable for such simula-tions. First, since the vorticity is differenced conservatively, the method preservesthe total circulation of the flow, i.e., the sum of the discrete vorticity does notchange in time (it is in fact zero for all the periodic problems studied here). Second,the method is designed to be ‘‘free stream preserving’’ which means that regionsof constant vorticity will remain constant as they are advected. Also, the non-centered differencing of the advective derivative terms makes this method robustto under-resolution, although the averaging procedure used to compute MAC-staggered velocities in Eq. (14) does add some amount of intrinsic diffusion tothe method.

3.7. Time Step Control

All of the methods mentioned above use some type of explicit calculation ofadvective derivatives that necessitates that the time step by limited by a CFLcondition. In practice each method recomputes an appropriate time step based onthe magnitude of the velocity at each step.

For all but the Godunov projection method the use of a Runge–Kutta temporalintegrator allows the time step to be limited by

Dt , Ch

maxi, j

(uui, ju 1 uvi, ju) 1 2n/h. (16)

In the computations presented, C is given the value 1.8 for the centered vorticitystream-function and pressure–Poisson method. A value of 0.9 is used for both theENO method and the upwind vorticity stream-function method. (See [28] for stabil-ity limits of TVD Runge–Kutta methods.) For the pseudospectral method the samecriterion as above is used with C 5 1.8/f [14] despite the fact that the implicittreatment of the viscous terms can increase the stability region.


For the Godunov projection method, the analysis in [24] suggests that

Dt , Ch/maxi, j

(uui, ju, uvi, ju) (17)

with C 5 1 is sufficient for stability. Here Eq. (17) is used with C 5 0.9 as the time-step control.

4. CONVERGENCE STUDIES OF THE NUMERICAL METHODS

In this section results of convergence studies for each of the methods describedin the last section are presented. The first of these will demonstrate each of themethods converging to the exact solution of a simple model problem as the meshspacing is reduced. The second will show convergence of the methods to a highlyaccurate reference solution of a shear layer problem. Later these results will becontrasted with those obtained from under-resolved simulations.

First a very simple flow is considered for which the exact solution is known.Taking as the domain the doubly-periodic unit square, the following is an exacttraveling wave solution of the Navier–Stokes equations

u(x, y, t) 5 1 1 2 cos(2f(x 2 t)) sin(2f(y 2 t))e28f2nt

v(x, y, t) 5 1 2 2 sin(2f(x 2 t)) cos(2f(y 2 t))e28f2nt

p(x, y, t) 5 2(cos(4f(x 2 t)) 1 cos(4f(y 2 t)))e216f2nt.

For each of the six numerical methods, the above problem is solved using a 2n 3

2n computational grid for n 5 4, 5, 6, and 7. The viscosity for each run was n 5

0.01. For each case the L1 norm of the error in the u-component of the velocitywas computed at time t 5 0.7, and the results are shown in Fig. 1. The convergencerates quoted in the caption of the figure were computed by taking the log2 of theratio of the errors from the 64 3 64 and 128 3 128 computations.

A few observations can be made from Fig. 1. The pseudospectral method is byfar the most accurate, but this is not surprising since the solution here consists ofa single Fourier mode. The fourth-order accuracy of the computed solution isinherited from the accuracy of the time integrator since the spatial accuracy is, ina sense, exact. This is in contrast to the primitive variable ENO method wherefourth-order convergence is exhibited despite the method having only third-orderaccuracy in time. The fact that for this problem the overall error is dominated bythe spatial error has nothing to do with the ENO type of discretization; the samebehavior is shared by the centered pressure–Poisson method if a less accuratetemporal integration method is used. Both of these methods appear slightly lessaccurate than the centered vorticity stream-function method, a fact probably dueto the use of the exact spectral representation of the inverse Laplacian used in thelatter. Finally, the two second-order methods produce nearly identical results despitethe third-order time integrator of the upwind vorticity stream-function method.Again this is due to the fact that spatial errors dominate the total error, and bothhave second-order accuracy in space.


FIG. 1. Convergence results for the traveling wave example. The computed convergence rates are(from top to bottom) Godunov projection, 2.04; upwind vorticity stream-function, 2.08; pressure–Poisson,3.99; ENO, 4.07; centered vorticity stream-function, 4.01; and pseudospectral, 4.00.

The second convergence test presented uses a problem involving a doubly periodicpair of shear layers. This is the ‘‘thick’’ shear layer problem from [9] with initialconditions given on the periodic unit square

u 5 Htanh(r(y 2 0.25)), for y # 0.5

tanh(r(0.75 2 y)), for y . 0.5(18)

v 5 d sin(2f(x 1 0.25)),

where r is the shear layer width parameter and d the strength of the initial perturba-tion. Since an exact solution is not known for this problem, a reference solutioncomputed using a very fine grid was used as the ‘‘exact’’ solution for determiningerrors in the convergence tests. For the examples to be discussed here, the referencesolution was computed to time t 5 1.0 using the pseudospectral method on a768 3 768 grid with r 5 30, d 5 0.05, and the viscosity n 5 0.002. The grid waschosen to be finer than necessary to resolve all physically relevant scales; this canbe monitored by examining the spectrum of the flow. Using the highest-ordermethod available assures that the error in the reference solution will be as smallas possible.

Each method was used to compute the solution of the thick shear layer problemto time t 5 1.0 using 64 3 64, 128 3 128, and 256 3 256 grids. The convergencerate of each method was estimated in two ways, by comparison with the referencesolution and by a procedure based on Richardson extrapolation.


FIG. 2. Convergence results for the thick shear layer example. Shown are results for each methodon meshes of size 64 3 64, 128 3 128 and 256 3 256. The computed convergence rates are (from topto bottom) upwind vorticity stream-function, 2.44; Godunov projection, 2.19; ENO, 3.90; pressure–Poisson, 3.92; centered vorticity stream-function, 3.91; and pseudospectral, 11.27.

Figure 2 displays the L1 errors for each run computed at t 5 1.0 by comparisonwith the reference solution. Table I shows the estimated convergence rates for thesame runs. The bottom two rows of this table are calculated by comparing computedsolutions to the reference solution and correspond to the slope of the line segmentsin Fig. 2. In the Richardson extrapolation procedure, the L1 diference between the64 3 64 and 128 3 128 solutions and the L1 difference between the 128 3 128 and256 3 256 solutions are calculated. These quantities are listed in the first two rows

TABLE IEstimated Errors and Convergence Rates from the Thick Shear Layer Example. (See

Also Fig. 2.) (PSpect 5 Pseudo-spectral, VSCent 5 Centered Vorticity Stream-Function,ENO 5 Primitive Variable ENO, PPois 5 Pressure-Poisson, VSUp 5 Upwind VorticityStream-Function, GProj 5 Godunov Projection.)

PSpect VSCent ENO PPois VSUp GProj

64–128 errors 1.05e-3 3.20e-3 1.17e-2 2.60e-2 1.40e-2 5.69e-3128–256 errors 2.87e-6 3.15e-4 9.61e-4 2.65e-3 2.65e-3 8.35e-4Richardson rate 8.09 3.35 3.61 3.30 2.35 2.7764–128 rate 9.75 3.39 3.65 3.38 2.39 2.68128–256 rate 11.27 3.91 3.90 3.92 2.44 2.19


of Table I and should be a reasonable approximation to the errors in the 64 3 64and 128 3 128 runs, respectively. Taking the log2 of the ratio of these quantitiesgives the estimated convergence rate that is listed in the row labeled ‘‘Richardsonrate.’’ This should be a reasonable estimate to the convergence rate between 64 3

64 and 128 3 128 runs, and Table I shows this to be valid. The agreement between theconvergence rates estimated for the pseudospectral method using both Richardsonextrapolation and comparison with the reference solution further substantiate theclaim that it is reasonable to take the 768 3 768 pseudospectral calculation as arepresentation of the exact solution. Indeed, the curve for the pseudospectralmethod in Fig. 2 suggests that the error in the reference solution is substantiallyless than 1029.

5. THIN SHEAR LAYERS

To illustrate the performance of the numerical methods on an under-resolvedflow, each method is used to compute the evolution of a thin shear layer. The initialconditions are given by Eq. (18) with r 5 80 and d 5 0.05, and the viscosity hereis 0.0001. These initial conditions are nearly the same as the thin shear layer problemin [9], but with a slightly thicker shear profile. Note that while we characterize thisshear layer as ‘‘thin’’ the transition between states in the shear layer occurs overapproximately six mesh points on a 128 3 128 mesh.

5.1. Appearance of Spurious Vortices

The above problem was computed using each method on both a 256 3 256 anda 128 3 128 grid. Figures 3 and 4 show vorticity contours from these runs at timet 5 0.6 and t 5 1.0, respectively. For each method and time, the plot displayscontours from the 256 3 256 run on the top portion of the plot and contours fromthe 128 3 128 on the bottom. (In reality, the 256 3 256 data is averaged andsuperimposed onto the coarser grid before plotting.) Since the problem has atop–bottom symmetry, it is sufficient to view only one shear layer from each calcula-tion. The three frames on the right side of each figure show the results for the threerobust methods considered while those on the left show results from the threenonrobust methods.

The results from the 256 3 256 runs appear nearly identical. This is a goodindication that all of the methods are capable of computing a reasonably accuratesolution, given sufficient resolution. There are, however, substantial differences inthe 128 3 128 runs. Note that the robust methods all produce an additional spuriousvortex in the shear layer midway between the main vortices. The nonrobust methods,while they do not produce spurious vortices at this resolution, clearly show thebeginnings of unstable oscillations, particularly for t 5 1.0. As the flow developsfurther, the latter methods will eventually experience catastrophic blowup as thesolution becomes more and more under-resolved.


FIG. 3. Contour plots of vorticity from thin shear layer runs at time t 5 0.6. In each of the sixframes, results from the 256 3 256 computation are shown in the top half of the plot, while resultsfrom the 128 3 128 are shown on the bottom.

5.2. Convergence Tests

In Section 4 we demonstrated numerical convergence for each of our methodson a thick shear layer problem. Here it will be shown how these same tests breakdown in the under-resolved case.

The two convergence study procedures used earlier for the thick shear layerexample were repeated using the data from the thin shear layer runs (r 5 80, d 5

0.05, and n 5 0.0001) computed at t 5 1.0. The results are shown in Table II.


FIG. 4. Contour plots of vorticity from thin shear layer runs at time t 5 1.0.

Several points should be made. First, it is clear that the convergence rate estimatesbased on the Richardson procedure no longer indicate that the methods are converg-ing to the proper order of accuracy. These numbers should be reasonable approxima-tions to the convergence rates computed with the 64 3 64 and 128 3 128 runs.Here these approximations are not nearly as accurate as in the thick shear layercase, but still give a reasonable indication of how badly convergence is failing.Second, the convergence rates listed in the last row for the upwind vorticity stream-function and Godunov methods are inflated by the fact that the 128 3 128 runsfor these methods contain prominent spurious vortices while the 256 3 256 runs


TABLE IIEstimated Errors and Convergence Rates from the Thin Shear Layer Example.

Labels as in Table I

PSpect VSCent ENO PPois VSUp GProj

64–128 errors 9.47e-2 3.04e-2 8.26e-2 3.79e-2 7.22e-2 9.27e-2128–256 errors 1.06e-2 3.06e-3 1.05e-1 1.19e-2 6.63e-2 7.45e-2Richardson rate 3.15 3.31 20.35 1.66 0.12 0.3164–128 rate 2.77 1.54 0.48 1.39 0.86 0.97128–256 rate 11.58 3.63 1.69 3.63 2.53 3.83

do not. Also, hyperviscosity was needed to stabilize the 64 3 64 run for the pressure–Poisson method which may somewhat affect the convergence rates for this method.

As was pointed out in the [9], it may be difficult to determine from gross statisticswhether spurious vortices have appeared in a computation. To demonstrate this,Figs. 5 and 6 display the computed energy and enstrophy of the solutions for thedifferent methods for the thin shear layer problem on a 128 3 128 grid. The ‘‘exact’’data, taken from the run used as the reference solution in the convergence tests,are superimposed with open circles on the plot. It is clear that these plots are not

FIG. 5. Energy decay for the thin shear layer example on a 128 3 128 grid. The data from the‘‘exact’’ solution are represented by open circles.


FIG. 6. Enstrophy decay for the thin shear layer example on a 128 3 128 grid. The data from the‘‘exact’’ solution are represented by open circles.

diagnostic when trying to determine whether physically reasonable solutions arebeing computed. For example, while the plots in Fig. 4 of the ENO method clearlyshow the effects of under-resolution, the energy and enstrophy curves in Figs. 5and 6 for the ENO method lie as close to the exact curves as do those for any ofthe methods and so would not be a cause for suspicion if viewed independently ofany other evidence.

It is also important to note that despite the under-resolved nature of the flows,all of the methods add little numerical diffusion to flow, especially in the earlystages of the computation. In other words, it is the physical viscosity that is thedominant diffusive mechanism for these runs. A sufficient amount of artificialviscosity will, in fact, prevent spurious vortices (see, e.g., [20]), but at the expenseof increasing the width of the underlying shear layers. The mechanism for this isclear; adding too much artificial viscosity can increase the smallest length scale ofthe problem to the point that it can be accurately represented on the computationalgrid, suppressing the initiation of such unstable artifacts.

5.3. Spurious Vortices for the Centered Methods

The examples of the previous section might give the impression that the nonrobustnumerical methods will not produce spurious vortices. However, these methodswill also produce incorrect solutions, but on coarser grids or for thinner shear layers


than the robust methods. In this section numerical experiments using very thinshear layers are presented for which this happens.

The pseudospectral, centered vorticity stream-function, and pressure–Poissonmethods were run on problems using thinner shear layers on a 128 3 128 grid.Specifically, for the pseudospectral method, a shear width parameter of r 5 100and viscosity n 5 0.0001 were used. For the vorticity stream-function method r 5

120 and n 5 0.00005, and for the pressure–Poisson method r 5 150 and n 5 0.0001.Hyperviscosity is used to stabilize the pressure–Poisson method here; hence athinner shear layer was required than for the other two methods to produce similarresults. Contour plots of the vorticity at time t 5 0.6 for these three runs are shownin the left column of Fig. 7 and at time t 5 1.0 in Fig. 8. The contour levels forthese plots are arranged so that the zero level of vorticity is not plotted, so thesubstantial amount of small scale oscillation for these runs is not visible.

The left-hand column of these figures clearly shows the appearance of spuriousvortices in the shear layers as in the last example, but for the pseudospectral andvorticity stream-function methods, two vortices rather than one appear in eachshear layer. Remarkably, if the initial conditions are shifted a half grid cell in thevertical direction, these two methods produce one extra vortex, but the pressure–Poisson method produces two. These runs are shown in the right-hand columns ofFigs. 7 and 8. The tendency to produce two versus one spurious vortex appears tobe linked to the use of vorticity in the calculation of the advective derivatives. Thisspeculation is supported by the discussion in Section 6.2 and is related to thecentering of the initial conditions with respect to the mesh. For the plots in theright-hand column, the zero-line of the vorticity in the initial shear layer lies alonga horizontal row of discretization points. For the plots in the left-hand column, thezero-line lies between two rows of discretization points.

6. SPECULATION ON THE SPURIOUS VORTEX PRODUCTION MECHANISM

It might seem somewhat remarkable that, under conditions of insufficient resolu-tion, all six of the methods presented here break down in the same way, by producingspurious vortices. However, this does suggest that there is an underlying mechanismcommon to all methods which is responsible for this phenomenon. We discuss sucha mechanism in this section.

6.1. Perturbed Initial Conditions

For the continuous case, strong shear layers such as those used in the abovestudy are unstable to perturbations in a range of wave numbers. The extent of thisrange depends on the viscosity of the flow and the thickness of the shear layer (see,e.g., [3, 11]). For the above examples, the shear layer was perturnbed using thelowest available wave number, but higher-frequency perturbations will also causethe shear layers to roll up. Realizing this, we can add a second wavenumber perturba-tion to the initial conditions above and produce resolved results that look verysimilar to the spurious vortex results.


Specifically, consider the perturbation flow given by

u 5 « cos(4fx 1 0.25) cos(2fy)(19)

v 5 2« sin(4fx 1 0.25) sin(2fy).

When added to the basic perturbed shear layer flow (18), this function perturbsthe vertical velocity with wavenumber two, i.e., twice the frequency of the originalperturbation. Using the value « 5 0.00375, the perturbation in (19) was added to

FIG. 7. Contour plots of vorticity from nonrobust methods on very thin shear layers runs at timet 5 0.6.


FIG. 8. Contour plots of vorticity from nonrobust methods on very thin shear layers runs at timet 5 1.0.

the initial conditions (18) using the thin shear layer values r 5 80, d 5 0.05, andn 5 0.0001. The resulting initial value problem was then solved using the centeredvorticity stream-function method on a 512 3 512 grid. The results are shown inFig. 9. Note that here the extra vortices in the middle of the shear layers are notthe result of under-resolution, but are real features of the flow.

This numerical experiment holds the key to understanding the tendency for allof the numerical methods to produce similar results when under-resolving flows.As is well known (see, e.g., [14]), the computational error due to discretization of


a differential equation satisfies the corresponding linearized difference equationforced by the truncation error for the method. By Duhamel’s principle, a forcedlinear problem is equivalent to a superposition of initial value problems for thehomogeneous difference equation. Therefore we would expect that if the truncationerror contained perturbations similar to (19), then additional vortices might form.For the Navier–Stokes equations, the quadratic nonlinearity in the convective termswill produce a truncation error that is quadratic in derivatives of the exact solution.By appealing to Fourier analysis, we realize that the truncation error will containFourier modes at double the frequencies of those that appear in the underlyingflow. We therefore speculate that the appearance of the spurious vortices in manyof the calculations above is due to truncation error terms that perturb the solutionat the second or higher wavenumber. This argument also suggests that in particularlyunder-resolved calculations, one would expect to see additional spurious vorticesat wavenumber four and higher, which in fact was reported in [9] (see, e.g., Fig. 8).

The speculation that terms with wavenumber two in the truncation error ofthe numerical methods are responsible for the production of spurious vortices isconsistent with several key points of our numerical experiments. Most obviously,as the grid is refined, the size of the truncation error decreases, and the spuriousvortices in those calculations are less prominent or nonexistent. Also, the fact thatit is a low wavenumber feature of the flow that is causing the spurious vorticesexplains why the use of filtering techniques, such as the exponential spectral filtersin the primitive variable ENO method or hyperviscosity in the pressure–Poissonmethod, does little to affect the formation of spurious vortices. (As we remarkedin Section 5.2, however, the use of a sufficiently large amount of artificial viscositywill suppress the spurious vortices, but the width of the shear layer will be increasedas well, which in essence defeats the purpose of a ‘‘high-resolution’’ method.)

FIG. 9. Highly resolved evolution of perturbed shear layer on a 512 3 512 grid. Note that in thiscase the additional roll up in the shear layers is not the result of under-resolution.


If our speculation is correct, it should be the case that higher-order accuratemethods are less likely to produce spurious vortices than lower-order methods.This is substantiated by the fact that, for instance, reducing the accuracy of thepressure–Poisson method to second order results in a method with a slightly greatertendency to produce spurious vortices. However, we have also observed that therobust methods, even those of higher than second-order accuracy, have a greatertendency to produce spurious vortices than the nonrobust methods. This might beexplained partially by the fact that the centering of the numerical differentiationfor the two types of methods is different. The robust methods utilize noncenteredor upwind differencing near points of under-resolution, while the nonrobust meth-ods are all centered. For methods of the two types that are of the same order, thecoefficient of the leading order truncation error terms will be smaller for a centeredmethod than a noncentered one. We have observed however, that a (centered)second-order version of the pressure–Poisson method is still less likely to producespurious artifacts than is the primitive variable ENO method, which is fourth-orderaccurate in space. It therefore appears that this truncation error argument does notentirely explain the behavior of these methods.

As an additional comment, note that the form of the time differencing usedappears to have little to do with the production of the spurious vortices. As anexperiment, the ENO differentiation method in the ENO method was replacedwith centered differences of the same order. The result is a method that performsvery similarly to the pressure–Poisson method (i.e., resists forming spurious vorti-ces), despite the fact that it is the same order in time as the original ENO methodwhich readily produces spurious vortices. This also suggests that more elaborateENO methods such as biased or weighted ENO [21, 26] would produce betterresults than the method used here.

There remains another puzzling result from our experiments, namely, why thepseudospectral method, which is more accurate than any other method tested andis also a centered method, produces spurious vortices so readily. We discuss thisin the next section.

6.2. Truncation Error of the Pseudospectral Method

The fact that the pseudospectral method so readily produces spurious vorticeswas initially quite puzzling. After all, it appears to be the most accurate of all themethods considered and should therefore resolve the low wavenumbers of the flowmuch better than other methods. The explanation for this lies in the formulationof the incompressible Navier–Stokes equations that is used for this method andthe particular model problem considered.

Recall from Section 3.2 that the nonlinear terms for the pseudospectral methodare based on a projection of the quantity g 3 U. This contains the term uuy whichfor the thin shear layer initial conditions is quite unsmooth. In the method, theterms are formed by first computing the vorticity in spectral space, and then formingthe quantity g 3 U in real space. This quantity is then transformed back to spectralspace and used in the temporal integration. To examine the aliasing error involvedin this procedure, we simply mimic the numerical method and compute the spectral


representation of uuy for the thin shear layer initial conditions with r 5 100 usingonly 128 points in the vertical. Figure 10 shows the magnitude in the first 15wavenumbers of this error. Because of the symmetry of the shear layer, the errorfirst appears in the 4f mode, i.e., the same mode responsible for creating an addi-tional rollup in the shear layer for the other nonspectral methods.

This argument is further substantiated by performing an experiment in whichthe usual form of the advective terms in the primitive variable formulation (U ? =)Uis used instead of the rotational form g 3 U. This formulation has the effect ofavoiding the aliasing error of the uuy term. The resulting method is a pseudospectralversion of the standard projection method. In this form, instead of a uuy termappearing in the equation for v, a vuy term appears in the equation for u. Numericaltests confirm that this formulation can resolve our shear layer problem withoutproducing spurious vortices up to about r 5 150 on a 128 3 128 grid. This changemakes the method very similar to the pressure–Poisson method, and this is evidentin numerical experiments, even to the point of producing one spurious vortex asopposed to two and two as opposed to one when the data is shifted a half cell inthe vertical as per the discussion in Section 5.3.

The differences in the performance of the two versions of the pseudospectralmethod seem to disappear if the initial conditions of the problem are rotated withrespect to the computational grid. In this case, both partial derivatives of bothvelocity components are unsmooth. Rotating the initial conditions also removes

FIG. 10. Aliasing error in the computation of uuy at t 5 0 for a 128 3 128 grid.


FIG. 11. Contour plots of vorticity with shear layer initial conditions rotated with respect to thecomputational grid.

any reason to believe that spurious vortices are only produced when the initialshear layers are parallel to the grid axis. Figure 11 shows results obtained with theGodunov projection method using shear layer initial conditions which have beenrotated by an angle equal to arctan(1/2). A single spurious vortex is clearly evidentin each of the shear layers, and in fact double spurious vortices can be producedby shifting the initial conditions. Similar results can be produced with the othermethods as well.

7. ADAPTIVE MESH REFINEMENT

In the design and implementation of numerical methods that use adaptive meshrefinement techniques it is necessary to devise criteria for determining when andwhere the computational mesh should be refined. In general, some sort of errorestimation is used, and the mesh is refined in regions where the error is unacceptablylarge. Both Richard-extrapolation-based procedures and strategies based on moni-toring the local magnitude of flow quantities are common and are discussed in theliterature [6–8]. In the context of the present paper, error estimation is essentiallyequivalent to deciding when and where a numerical method is under-resolving theflow. Additionally, as a time-dependent computation proceeds, grid refinementmust occur before nonphysical effects such as spurious vortices appear in the flow.There is therefore an intimate connection between studying the effects of under-resolution and developing effective strategies for AMR. In this section some simplenumerical experiments are presented that are designed to emphasize some of thekey concerns relevant to using AMR for incompressible flow.


7.1. A Projection Method for Locally Refined Grids

The numerical method used for the AMR examples to follow is taken from theprojection method for locally refined grids presented in [25]. This method uses ablock structured grid refinement strategy similar to that originally described byBerger and Oliger [7, 8]. Within this grid hierarchy, the Godunov projection methoddescribed in Section 3 is implemented. Each grid patch is updated every time stepregardless of cell size using boundary conditions which are either interpolated fromcoarser grids or given by the physical boundary conditions (here periodicity). Thisis in contrast to other methods in which temporal as well as spatial refinement isused [1, 29]. This organization has the advantage that for each of the explicitoperations performed in the method, the AMR method for a single grid patch usesthe same procedures as a single-grid method. The exception to this design philosophycomes when the implicit parts of the method are implemented.

The Poisson equation associated with the projection step in this method is globalin nature and is solved on the entire AMR structure simultaneously. The multigridtechnique used, wherein relaxation on individual grid patches is combined withcoarsening and interpolation sweeps through the AMR structure, is described in[25]. The method presented in [25] is for Euler flow only and has no descriptionof the solution of the elliptic equation associated with the Crank–Nicolson treatmentof the viscous terms. It is, however, straightforward to modify the multigrid methodfor the Poisson equation to solve this slightly different elliptic equation, and thisis what is done here.

7.2. Local Refinement and Spurious Vortices

In this section, three calculations performed with the Godunov projection methodare presented in which AMR is used in an attempt to resolve the perturbed shearlayer flow problem and which illustrate the importance of a careful and effectiveerror estimation technique. In all three examples, an area of the computationalgrid near the point where spurious vortices are known to occur is refined by afactor of two over the base grid. The refinement is present at the initial time t 5

0, and the refined regions do not change during the calculations. The difference inthe three experiments is the extent of the refined region and the resolution of thebase grid.

In the first two examples, a square refinement patch with sides of length 0.25 iscentered over the point where a spurious vortex is known to appear. The firstexample uses a 128 3 128 base grid so that the effective resolution in the refinedpatch is that of a 256 3 256 grid. The computed solution at time t 5 0.8 is shownin Fig. 12. The location of the refinement patch is indicated by outlining it on thecontour plot.

For the same method, a 256 3 256 grid is known to not produce a spuriousvortex, and so since the refinement patch has the same local resolution as thatgrid, one might expect the resulting computation not to produce such an artifact.However, as is apparent in the figure, this simple attempt at mesh refinement isunsuccessful in suppressing the spurious vortex. The source of the problem has todo with the fact that a factor-of-two refinement occurs in the middle of a region


FIG. 12. Contour plots of vorticity with naive mesh refinement.

(i.e., the region around the shear layer) that is not sufficiently resolved on the128 3 128 mesh. As a result the shear layer is not represented smoothly across theinterface of the coarse and fine meshes, and the spurious vortex is initiated. Thesame experiment is repeated using a highly resolved 512 3 512 base grid (givingan effective resolution of 1024 3 1024 grid), and the results are displayed in Fig.13. Here the naive grid refinement at least does no damage to the solution up tothe end of the calculation at time t 5 1.0.

FIG. 13. Contour plots of vorticity with naive mesh refinement but with a resolved coarse grid so-lution.


FIG. 14. Contour plots of vorticity using AMR with the entire upper shear layer refined.

Figure 14 shows results from a computation in which the refinement region isinitially arranged so that the entire upper shear layer will be contained in the refinedregion throughout the calculation. A 128 3 128 base grid is used giving the effectiveresolution of a 256 3 256 grid for the upper shear layer. Comparing the upper andlower shear layers, it is seen that the use of grid refinement has prevented thespurious vortex from appearing in the upper shear layer without affecting the lowershear layer.

The lesson learned from this simple comparison is that an AMR strategy forlocal refinement must be done intelligently. Grid interfaces between coarse andfine grids should not lie across unstable structures unless these structures are fullyresolved on the coarse grid. Also, if refinement is to be increased as a computationevolves and smaller scales develop, grid refinement must be employed before largescale effects of under-resolution (e.g., spurious vortices) have occurred. Any proce-dure for automatically determining where a computational grid should be refinedmust address these issues.

8. DISCUSSION

In this paper, we have extended our earlier work on under-resolution in incom-pressible flow calculations [9] by presenting further numerical experiments using aset of upwind and centered algorithms that is representative of many of the currentlypopular finite difference algorithms for approximating the incompressible Navier–Stokes equations. It has been demonstrated that the production of spurious vorticesin the double shear layer problem is a result of insufficient resolution and occurswith both upwind and centered type methods. These artifacts are not a high-wavenumber effect, but rather represent the growth of unstable low-wavenumber


perturbations introduced by the truncation error of the methods. Since these spuri-ous vortices appear as large smooth structures in a computation, it is difficult todifferentiate them from the physical features of the flow. It may not be possible toboth prevent these spurious vortices by using artificial viscosity or filtering tech-niques and to maintain the correct physical smallest length scales in a computation.Adding a sufficient amount of artificial viscosity will, in fact, prevent these artifacts,but at the expense of increasing the width of the underlying shear layers. Thissubverts our stated objective, which is to compute solutions to the Navier–Stokesequations in which small length-scale features are represented with as close to thecorrect scales as can be supported by the computational mesh.

Spurious vortices can be prevented for all methods by computing on a grid thatis sufficiently fine to resolve the smallest scales in the problem of interest. Theresolution required clearly depends on the difference method. For example, wehave observed that in general centered methods require fewer grid points than doupwind methods in order to prevent these artifacts. It is, not surprisingly, also thecase that higher-order accurate methods of a given class require less resolutionthan do lower-order methods. While this might seem to constitute an endorsementof centered methods, one must keep in mind that in general, it is difficult to stabilizecentered methods in the nearly under-resolved case using artificial viscosity, withoutat the same time swamping the physical viscosity, and hence destroying the desiredsmallest length-scale resolution.

Choosing a mesh size to optimally resolve the smallest length scales in a problemusually requires a careful mesh convergence study. As we have shown, in theabsence of spurious vortices, all of the methods considered here converge at theexpected rates as the mesh is refined. As a general strategy, making a mesh refine-ment study for every computation may not be practical. A more efficient alternativeis to use adaptive mesh refinement to increase resolution only where necessary.We have given brief evidence in this paper that a careful AMR strategy will preventspurious vortices. Using a procedure based on Richardson extrapolation for errorestimation may be an effective strategy for determining where to refine the mesh,although monitoring the local magnitude of flow quantities such as the vorticitymay also be sufficient. The question of how to implement effective error estimationcriteria for this problem is a topic for future research.

ACKNOWLEDGMENTS

The authors thank Weinan E and Chi-Wang Shu for providing us with a version of the ENO methodused in this work and Michael Shelley for providing a prototype of the pseudospectral code andmany helpful comments. We also thank Ricardo Cortez and Bill Henshaw for helpful advice andcareful proofreading.

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